Which Is The Best First Step When Solving The Following System Of Equations?$\[ \begin{align} y &= 7x + 3 \\ y &= 9x \end{align} \\]A. Multiply The First Equation By -7.B. Substitute \[$9x\$\] For \[$y\$\] In The First

Feb 28, 2025 by ADMIN 221 views

**Which is the Best First Step When Solving a System of Equations?**

Introduction

Solving a system of equations is a fundamental concept in mathematics, and it is essential to understand the best approach to tackle such problems. In this article, we will explore the best first step when solving a system of equations, using the given system of equations as an example.

Understanding the System of Equations

The given system of equations is:

{ \begin{align*} y &= 7x + 3 \\ y &= 9x \end{align*} \}

In this system, we have two equations with two variables, x and y. The first equation is y = 7x + 3, and the second equation is y = 9x. Our goal is to find the values of x and y that satisfy both equations.

Option A: Multiply the First Equation by -7

One possible approach to solving this system of equations is to multiply the first equation by -7. This would give us:

{ \begin{align*} -7y &= -49x - 21 \\ y &= 9x \end{align*} \}

By multiplying the first equation by -7, we have created a new equation that is equivalent to the original first equation. However, this approach does not seem to be the most efficient way to solve the system of equations.

Option B: Substitute 9x for y in the First Equation

Another possible approach to solving this system of equations is to substitute 9x for y in the first equation. This would give us:

{ \begin{align*} 9x &= 7x + 3 \\ 9x - 7x &= 3 \\ 2x &= 3 \\ x &= \frac{3}{2} \end{align*} \}

By substituting 9x for y in the first equation, we have created a new equation that is equivalent to the original first equation. This approach seems to be more efficient than multiplying the first equation by -7.

Why Substitution is the Best First Step

Substitution is the best first step when solving a system of equations because it allows us to eliminate one of the variables and create a new equation that is equivalent to the original equation. In this case, substituting 9x for y in the first equation allowed us to eliminate the variable y and create a new equation that is equivalent to the original first equation.

Advantages of Substitution

Substitution has several advantages when solving a system of equations. Some of the advantages of substitution include:

Elimination of variables: Substitution allows us to eliminate one of the variables and create a new equation that is equivalent to the original equation.
Simplification of equations: Substitution can simplify the equations and make them easier to solve.
Reduced number of variables: Substitution can reduce the number of variables in the system of equations, making it easier to solve.

Conclusion

In conclusion, the best first step when solving a system of equations is to substitute one of the variables in one of the equations. In this case, substituting 9x for y in the first equation allowed us to eliminate the variable y and create a new equation that is equivalent to the original first equation. Substitution has several advantages when solving a system of equations, including the elimination of variables, simplification of equations, and reduction of the number of variables.

Real-World Applications

Solving a system of equations has several real-world applications, including:

Physics and engineering: Solving a system of equations is essential in physics and engineering, where it is used to model real-world problems and make predictions.
Economics: Solving a system of equations is used in economics to model economic systems and make predictions about economic trends.
Computer science: Solving a system of equations is used in computer science to model complex systems and make predictions about their behavior.

Final Thoughts

In conclusion, solving a system of equations is a fundamental concept in mathematics, and it is essential to understand the best approach to tackle such problems. Substitution is the best first step when solving a system of equations, as it allows us to eliminate one of the variables and create a new equation that is equivalent to the original equation. By understanding the best approach to solving a system of equations, we can apply this knowledge to real-world problems and make predictions about complex systems.

Additional Resources

For additional resources on solving a system of equations, including tutorials, examples, and practice problems, please visit the following websites:

Khan Academy: Khan Academy offers a comprehensive tutorial on solving a system of equations, including examples and practice problems.
Mathway: Mathway is an online math problem solver that can help you solve a system of equations and provide step-by-step solutions.
Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you solve a system of equations and provide step-by-step solutions.

References

Algebra: Algebra is a fundamental concept in mathematics that deals with the study of variables and their relationships.
Linear equations: Linear equations are a type of equation that can be written in the form ax + by = c, where a, b, and c are constants.
Systems of equations: Systems of equations are a set of two or more linear equations that are solved simultaneously to find the values of the variables.
Frequently Asked Questions (FAQs) About Solving a System of Equations ====================================================================

Q: What is a system of equations?

A: A system of equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables.

Q: Why do we need to solve a system of equations?

A: We need to solve a system of equations to find the values of the variables that satisfy all the equations in the system. This is useful in many real-world applications, such as physics, engineering, economics, and computer science.

Q: What are the different methods for solving a system of equations?

A: There are several methods for solving a system of equations, including:

Substitution method: This method involves substituting one of the variables in one of the equations with an expression from the other equation.
Elimination method: This method involves eliminating one of the variables by adding or subtracting the equations.
Graphical method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the substitution method?

A: The substitution method involves substituting one of the variables in one of the equations with an expression from the other equation. This is done to eliminate one of the variables and create a new equation that is equivalent to the original equation.

Q: What is the elimination method?

A: The elimination method involves eliminating one of the variables by adding or subtracting the equations. This is done to create a new equation that is equivalent to the original equation.

Q: What is the graphical method?

A: The graphical method involves graphing the equations on a coordinate plane and finding the point of intersection. This method is useful for systems of equations with two variables.

Q: How do I choose the best method for solving a system of equations?

A: The best method for solving a system of equations depends on the specific system and the variables involved. If the system has two variables, the graphical method may be the best choice. If the system has more than two variables, the substitution or elimination method may be more suitable.

Q: What are some common mistakes to avoid when solving a system of equations?

A: Some common mistakes to avoid when solving a system of equations include:

Not checking the solution: Make sure to check the solution to ensure that it satisfies all the equations in the system.
Not using the correct method: Choose the best method for solving the system of equations based on the specific system and variables involved.
Not simplifying the equations: Simplify the equations before solving the system to make it easier to find the solution.

Q: How do I check the solution to a system of equations?

A: To check the solution to a system of equations, substitute the values of the variables into each equation and check if the equation is true. If the equation is true, then the solution is correct.

Q: What are some real-world applications of solving a system of equations?

A: Solving a system of equations has many real-world applications, including:

Physics and engineering: Solving a system of equations is essential in physics and engineering, where it is used to model real-world problems and make predictions.
Economics: Solving a system of equations is used in economics to model economic systems and make predictions about economic trends.
Computer science: Solving a system of equations is used in computer science to model complex systems and make predictions about their behavior.

Q: How can I practice solving systems of equations?

A: You can practice solving systems of equations by working through examples and exercises in a textbook or online resource. You can also try solving systems of equations on your own using a calculator or computer program.

Q: What are some online resources for learning about solving systems of equations?

A: Some online resources for learning about solving systems of equations include:

Khan Academy: Khan Academy offers a comprehensive tutorial on solving systems of equations, including examples and practice problems.
Mathway: Mathway is an online math problem solver that can help you solve systems of equations and provide step-by-step solutions.
Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you solve systems of equations and provide step-by-step solutions.